Write an exponential function π¦=πππ₯for a graph that passes through (2,1) and (3,10).
y = ab^x
ab^2 = 1
ab^3 = 10
divide and you have b = 10
a*100=1
y = 1/100 * 10^x
or, though not in the form requested,
y = 10^(x-2)
it was kind of a cop-out to have a be a power of b.
To write an exponential function in the form π¦=πππ₯, where π represents the initial value and π represents the base, we will first determine the values of π and π using the given points (2,1) and (3,10).
Let's start by substituting the coordinates (2,1) into the equation π¦=πππ₯:
1 = ππ^2 ---(1)
Next, substitute the coordinates (3,10) into the equation π¦=πππ₯:
10 = ππ^3 ---(2)
To solve for π and π, we'll divide equation (2) by equation (1):
10/1 = (ππ^3)/(ππ^2)
Simplifying this expression gives:
10 = π
Now, substitute this value of π into equation (1):
1 = π(10)^2
Simplifying further:
1 = 100π
Divide both sides by 100:
1/100 = π
So, we have found the values of π and π:
π = 1/100
π = 10
Therefore, the exponential function that passes through the points (2,1) and (3,10) is:
π¦ = (1/100)(10)^π₯
To write an exponential function in the form π¦ = πππ₯, we need to find the values of π and π.
Given that the graph passes through the points (2,1) and (3,10), we can use these values to find π and π.
Step 1: Substitute the first point, (2,1), into the equation π¦ = πππ₯:
1 = ππΒ²
Step 2: Substitute the second point, (3,10), into the equation π¦ = πππ₯:
10 = ππΒ³
Step 3: Divide the second equation by the first equation to eliminate π:
10 Γ· 1 = (ππΒ³) Γ· (ππΒ²)
10 = π
So, we have found the value of π, which is 10.
Step 4: Substitute the value of π back into either of the original equations to find π. Let's use the first equation:
1 = π(10)Β²
Simplifying:
1 = 100π
Dividing both sides by 100:
1/100 = π
So, we have found the value of π, which is 1/100.
Therefore, the exponential function that passes through the points (2,1) and (3,10) is:
π¦ = (1/100)(10)^π₯